# Integrating over the Y Axis: (∆CS)

The traditional formulae for consumer surplus is: $$\text{CS} = \int_{0}^{x_0} [x(p_x,\overline{p_y}, \overline{m})]dx - x_0P_{x_0}$$.

This is the area under the Marshallian demand curve, that is only the triangle above $$P_x$$

However I noticed that the formulae for the compensating variation, which is a sliver underneath the Hicksian demand curve ($$h$$) between $$P_{x_0}$$ and $$P_{x_1}$$ is calculated using $$\text{CS} = \int_{p_{x_0}}^{p_{x_1}}[h(p_x,\overline{p_y}, \overline{u})]dp_x$$

Questions:

1. Could we calculate ∆CS the same way i.e. $$\text{∆CS} = \int_{p_{x_0}}^{p_{x_1}} [x(p_x,\overline{p_y}, \overline{m})]dp_x$$ instead of using the lengthier: $$\text{∆CS} = [\int_{0}^{x_0} [x(p_x,\overline{p_y}, \overline{m})]dx - x_0P_{x_0}]$$ - $$[\int_{0}^{x_1} [x(p_x,\overline{p_y}, \overline{m})]dx - x_1P_{x_1}]$$

2. More generally can we chose to integrate with respect to the Y axis like this, when we want a sliver of the graph as per these examples?

3. If i can integrate with respect to the y axis is there anything I need to be careful of?

I have attached an image to give additional clarity to my question. Thank you!

1. The traditional formula for $$CS$$ would actually be $$CS = \int_{0}^{x_0} [p_x(x,\overline{p_y},\overline{m})] dx - p_0 x_0$$. In traditional integration with respect to the horizontal axis, we integrate the vertical axis variable with respect to the horizontal axis variable. This implies we need to get the (partial) inverse demand with respect to $$x$$, i.e. solve for $$p_x$$ as a function of $$x$$ and the other variables as fixed.

For example: $$x(p_x, p_y, m) = \frac{m p_y}{p_x} \implies x = \frac{m p_y}{p_x} \implies p_x = \frac{m p_y}{x} \implies p_x(x,p_y,m) = \frac{m p_y}{x}$$.

1. The change in consumer surplus can actually be calculated as $$\Delta CS = \int_{p_{x_0}}^{p_{x_1}} [x(p_x,\overline{p_y},\overline{m})] dp_x$$.

The lengthier version is wrong as it is based on the wrong traditional formula for $$CS$$. Also, if $$x_0$$ corresponds to the original point and $$x_1$$ to the point after the price change, the lengthier version should actually correspond to $$\Delta CS = CS(x_1) - CS(x_0)$$, rather than $$\Delta CS = CS(x_0) - CS(x_1)$$, as the latter one would flip the sign of the change.

1. We can integrate like that with respect to the $$y$$ axis as long as the integrand is the dependent variable $$x$$ as a function of $$y$$, i.e. $$x(y)$$.

To integrate with respect to the $$x$$ axis, we need to have $$y(x)$$.

1. What you need to be careful about is:
• Integrating with respect to the $$x$$ axis, if we had a function $$x(y)$$, we would need to solve for $$y$$ in terms of $$x$$, i.e. $$y(x)$$.
• Integrating with respect to the $$y$$ axis, if we had a function $$y(x)$$, we would need to solve for $$x$$ in terms of $$y$$, i.e. $$x(y)$$.
• Thank you very much for the detail Nicolas, I will go through tomorrow and then accept your answer once I have understood! On the brief skim I wasn't sure why my ∆CS calculation was wrong so I need to check that! Thanks for laying the answer out in such clarity! Commented Feb 16, 2023 at 4:14